Some comparison, symmetry and monotonicity results for Carnot-Carathéodory spaces

Abstract

We consider here some differential operators arising from the so called Carnot-Carathéodory metric spaces associated with a family of vector fields $X = (X_1, \ldots, X_k)$, which include the Hörmander type as a special case. We prove some weak and strong comparison results for solutions of the relevant differential $inequalities$. We then use these results to get some symmetry and monotonicity properties of solutions of the relevant partial differential equations.